What I mean by this is as follows:
Take an infinite flat manifold $\mathbb{R}^3$ with zero curvature. Then subtract out a knotted torus or linked tori. And sew them back in using Dehn surgery.
(In the 2 dimensional case, the resulting manifold will always have overall negative curvature. i.e. it is just a plane with cross-caps and handles.)
In the 3 dimensional case are there any manifolds made by this procedure that have overall zero or even positive curvature? Do you have any examples? And how would you work this out? Are there any that are "flat" as in zero-curvature at every point?
Another way to put this is are there any topologies which are $\mathbb{R}^3$ outside a certain region but have overall zero curvature? Or if it is impossible does this mean all infinite manifolds which are $\mathbb{R}^3$ outside a certain region and have overall zero curvature are equal to $\mathbb{R}^3$?
(The one I would try first would be similar to the way of getting the Poincare homology sphere by cutting out a trefoil knot in $\mathbb{R}^3$ and sewing it back in with +1 surgery. In fact you would only consider Dehn surgeries that give positively curved manifolds when done on $S^3$. But it doesn't prove that doing the same surgery on $\mathbb{R}^3$ would give a overall flat manifold.)
I suppose a third way of asking the same question is: "Are there any Dehn surgeries which don't alter the curvature of the original manifold".